Commutative version of the local Hamiltonian problem and common eigenspace problem

Abstract

We study the complexity of a problem Common Eigenspace --- verifying consistency of eigenvalue equations for composite quantum systems. The input of the problem is a family of pairwise commuting Hermitian operators H_1,\ldots,H_r on a Hilbert space (\CC^d)^{\otimes n} and a string of real numbers \lambda=(\lambda_1,\ldots,\lambda_r). The problem is to determine whether the common eigenspace specified by equalities H_a|\psi\ra=\lambda_a|\psi\ra, a=1,\ldots,r has a positive dimension. We consider two cases: (i) all operators H_a are k-local; (ii) all operators H_a are factorized. It can be easily shown that both problems belong to the class \QMA --- quantum analogue of \NP, and that some \NP-complete problems can be reduced to either (i) or (ii). A non-trivial question is whether the problems (i) or (ii) belong to \NP? We show that the answer is positive for some special values of k and d. Also we prove that the problem (ii) can be reduced to its special case, such that all operators H_a are factorized projectors and all \lambda_a=0.

Guest column

“…Just like in Ref. [BV13], the better the expanders (namely, the smaller δ is) the less appropriate the graphs are for NLTS.…”

“…However, this is done not by providing an explicit classical description of the groundstate, but rather in a much more indirect way. Very roughly, Schuch writes the Hamiltonian as the sum of two 2-local Hamiltonians; he can then use the Bravyi-Vyalyi machinery [BV13] with respect to each one of those, to construct a certificate which shows that the intersection of their two groundspaces is non-zero. Of course, if the qPCP conjecture is then the Hamiltonians in the construction cannot allow such indirect witnesses either.…”

“…For example, the main result of [BH13] can be viewed as ruling out Hamiltonians whose degree grows asymptotically with n from being good candidates for proving the NLTS conjecture. In this section we survey additional recent progress on this conjecture (both negative and positive) through several results [BV13,Has13,AE11,AE13a,Sch11,Has12]. All of those results apply to a special subclass of local Hamiltonians called commuting Hamiltonians, in which the terms of the Hamiltonian are required to mutually commute.…”

“…In the remainder of this section we restrict our attention to commuting local Hamiltonians (CLHs), in which all terms of the Hamiltonian mutually commute. The key statement that makes the situation in the case of CLHs simpler is a fundamental structural lemma due to Bravyi and Vyalyi (BV) [BV13], which we now describe. Although the original BV lemma was proven in the context of commuting 2-local Hamiltonians, we will then see how it can also be used to analyze k-local Hamiltonians.…”

“…Using this construction, Bravyi-Vyalyi [BV13] find an eigenbasis for H made of trivial eigenvectors, each one generated by a depth-2 quantum circuit. For example, a circuit generating a groundstate is defined as follows: acting on an initial product state, it first generates a (possibly entangled) state for each of the pairs of interacting sub-particles (as in Fig.…”